11 research outputs found
An Extremal Eigenvalue Problem for a Two-Phase Conductor in a Ball
The pioneering works of Murat and Tartar (Topics in the mathematical
modeling of composite materials. PNLDE 31. Birkhäuser, Basel, 1997) go a long
way in showing, in general, that problems of optimal design may not admit solutions
if microstructural designs are excluded from consideration. Therefore, assuming, tactilely,
that the problem of minimizing the first eigenvalue of a two-phase conducting
material with the conducting phases to be distributed in a fixed proportion in a given
domain has no true solution in general domains, Cox and Lipton only study conditions
for an optimal microstructural design (Cox and Lipton in Arch. Ration. Mech.
Anal. 136:101–117, 1996). Although, the problem in one dimension has a solution
(cf. Kre˘ın in AMS Transl. Ser. 2(1):163–187, 1955) and, in higher dimensions, the
problem set in a ball can be deduced to have a radially symmetric solution (cf. Alvino
et al. in Nonlinear Anal. TMA 13(2):185–220, 1989), these existence results have
been regarded so far as being exceptional owing to complete symmetry. It is still not
clear why the same problem in domains with partial symmetry should fail to have a
solution which does not develop microstructure and respecting the symmetry of the domain.We hope to revive interest in this question by giving a new proof of the result
in a ball using a simpler symmetrization result from Alvino and Trombetti
Spectral asymptotics of the Helmholtz model in fluid-solid structures
Artículo de publicación ISIA model representing the vibrations of a coupled fluid-solid structure is considered. This structure consists of a tube bundle immersed in a slightly compressible fluid. Assuming periodic distribution of tubes, this article describes the asymptotic nature of the vibration frequencies when the number of tubes is large. Our investigation shows that classical homogenization of the problem is not sufficient for this purpose. Indeed, our end result proves that the limit spectrum consists of three parts: the macro-part which comes from homogenization, the micro-part and the boundary layer part. The last two components are new. We describe in detail both macro- and micro-parts using the so-called Bloch wave homogenization method
Homogenization of a Transmission Problem in Solid Mechanics
Artículo de publicación ISIIn this paper we study a simplified model of the behavior of a 3-D solid made
from two elastic homogeneous materials separated by a rapidly oscillating interface.
We study the asymptotic beha¨ior of the solution of such model using
homogenization tools and a compactness result. We obtain the homogenized
equation, and by studying its coefficients, we find some properties of the limiting
material
Approximate controllability of a semilinear elliptic problem with robin condition in a periodically perforated domain
In this article, we study the approximate controllability and home-genization results of a semi-linear elliptic problem with Robin boundary condition in a periodically perforated domain. We prove the existence of minimal norm control using Lions constructive approach, which is based on Fenchel-Rockafeller duality theory, and by means of Zuazua's fixed point arguments. Then, as the homogenization parameter goes to zero, we link the limit of the optimal controls ( the limit of fixed point of the controllability problems) with the optimal control of the corresponding homogenized problem.SERB-DST Bhopal, YSS-2014-000732 / CREST, IISER Bhopal / CeBiB, PFBasal-01 / CMM, PFBasal-03 / Fondecyt,
1140773 / Regional Program STIC-AmSud Project Mosco
Numerical Maximization of the p-Laplacian Energy of a Two-Phase Material
For a diffusion problem modeled by thep-Laplacian operator, we are interested inobtaining numerically the two-phase material which maximizes the internal energy. We assume thatthe amount of the best material is limited. In the framework of a relaxed formulation, we presenttwo algorithms, a feasible directions method and an alternating minimization method. We show theconvergence for both of them, and we provide an estimate for the error. Since forp >2 both methodsare only well-defined for a finite-dimensional approximation, we also study the difference betweensolving the finite-dimensional and the infinite-dimensional problems. Although the error bounds forboth methods are similar, numerical experiments show that the alternating minimization methodworks better than the feasible directions one
Approximation of solutions to fractional integral equation
In this paper we shall study a fractional integral equation in an arbitrary Banach space X.
We used the analytic semigroups theory of linear operators and the fixed point method to
establish the existence and uniqueness of solutions of the given problem.Wealso prove the
existence of global solution. The existence and convergence of the Faedo Galerkin solution
to the given problem is also proved in a separable Hilbert space with some additional
assumptions on the operator A. Finally we give an example to illustrate the applications
of the abstract results.The first two authors would like to thank the CMM Santiago, University of Chile, for providing the financial support to
carry out this research work
A novel model for biofilm growth and its resolution by using the hybrid immersed interface-level set method
Artículo de publicación ISIIn this work we propose a new model to simulate biofilm structures (‘‘finger-like’’, as
well as, compact structures) as a result of microbial growth in different environmental
conditions. At the same time, the numerical method that we use in order to carry out the
computational simulations is new to the biological community, as far as we know. The use
of our model sheds light on the biological process of biofilm formation since it simulates
some central issues of biofilm growth: the pattern formation of heterogeneous structures,
such as finger-like structures, in a substrate-transport-limited regime, and the formation of
more compact structures, in a growth-limited-regime. The main advantage of our approach
is that we consider several of the most relevant aspects of biofilm modeling, particularly,
the existence and evolution of a biofilm–liquid interface. At the same time, in order to
perform numerical simulations, we have used sophisticated numerical techniques based on
mixing the immersed interface method and the level-set method, which are well described
in the present work.This work was partially supported by Millennium Scientific Initiative under grant number ICM P05-001-F. The work of
first author was also partially supported by Chilean Government Fondecyt-Conicyt Program under grant number 11080222
and by Universidad del Bío-Bío (grant DIUBB 121909 GI/C). Third author was also partially supported by Fondecyt grant
number 1130317
The Bloch Approximation in Periodically Perforated Media
We consider a periodically heterogeneous and perforated medium filling
an open domain of RN . Assuming that the size of the periodicity of the structure
and of the holes is O(ε), we study the asymptotic behavior, as ε → 0, of the solution
of an elliptic boundary value problem with strongly oscillating coefficients posed
in ε ( ε being minus the holes) with a Neumann condition on the boundary of
the holes. We use Bloch wave decomposition to introduce an approximation of the
solution in the energy norm which can be computed from the homogenized solution
and the first Bloch eigenfunction.We first consider the case where is RN and then
localize the problem for a bounded domain , considering a homogeneous Dirichlet
condition on the boundary of
Detecting a moving obstacle in an ideal fluid by a boundary measurement
In this Note we investigate the problem of the detection of a moving obstacle in a perfect fluid occupying a bounded domain in R-2 from the measurement of the velocity of the fluid on one part of the boundary. We show that when the obstacle is a ball, we may identify the position and the velocity of its center of mass from a single boundary measurement. Linear stability estimates are also established by using shape differentiation techniques.This work was achieved while the last author (LR) was visiting the Centro de Modelamiento Matemático at the
Universidad de Chile (UMI CNRS 2807). He thanks this institution for its hospitality, and the CNRS for its support.
The first author thanks the Millennium ICDB for partial support through grant ICM P05-001-F. The second author
was partially supported by CONICYT-FONDECYT grant 3070040. The authors also thank the Chilean and French
Governments through Ecos-Conicyt Grant C07 E05
On the graphene Hamiltonian operator
We solve a second-order elliptic equation with quasi-periodic boundary conditions defined on a honeycomb lattice that represents the arrangement of carbon atoms in graphene. Our results generalize those found by Kuchment and Post (Commun Math Phys 275(3):805-826, 2007) to characterize not only the stability but also the instability intervals of the solutions. This characterization is obtained from the solutions of the energy eigenvalue problem given by the lattice Hamiltonian. We employ tools of the one-dimensional Floquet theory and specify under which conditions the one-dimensional theory is applicable to the structure of graphene. The systematic study of such stability and instability regions provides a tool to understand the propagation properties and behavior of the electrons wavefunction in a hexagonal lattice, a key problem in graphene-based technologies.PFBasal-01 (CeBiB)
PFBasal-03 (CMM)
Comision Nacional de Investigacion Cientifica y Tecnologica (CONICYT)
C13E05
Comision Nacional de Investigacion Cientifica y Tecnologica (CONICYT)
CONICYT FONDECYT
1140773
1180781
Comision Nacional de Investigacion Cientifica y Tecnologica (CONICYT)
21110749
Spanish Government
SEV-2011-008